Download ∗-AUTONOMOUS CATEGORIES: ONCE MORE

Theory and Applications of Categories, Vol. , No. , , pp. 1–20.
∗-AUTONOMOUS CATEGORIES:
ONCE MORE AROUND THE TRACK
MICHAEL BARR
Transmitted by
To Jim Lambek on the occasion of his 75th birthday
ABSTRACT. This represents a new and more comprehensive approach to the ∗autonomous categories constructed in the monograph [Barr, 1979]. The main tool in
the new approach is the Chu construction. The main conclusion is that the category of
separated extensional Chu objects for certain kinds of equational categories is equivalent to two usually distinct subcategories of the categories of uniform algebras of those
categories.
1. Introduction
The monograph [Barr, 1979] was devoted to the investigation of ∗-autonomous categories.
Most of the book was devoted to the discovery of ∗-autonomous categories as full subcategories of seven different categories of uniform or topological algebras over concrete
categories that were either equational or reflective subcategories of equational categories.
The base categories were:
1. vector spaces over a discrete field;
2. vector spaces over the real or complex numbers;
3. modules over a ring with a dualizing module;
4. abelian groups;
5. modules over a cocommutative Hopf algebra;
6. sup semilattices;
7. Banach balls.
For definitions of the ones that are not familiar, see the individual sections below.
These categories have a number of properties in common as well as some important
differences. First, there are already known partial dualities, often involving topology.
This research has been supported by grants from the NSERC of Canada and the FCAR du Québec
AMS subject classification (1990): 18D10, 43A40, 46A70, 51A10.
Key words and phrases: duality, topological algebras, Chu categories.
c Michael Barr 1998. Permission to copy for private use granted.
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2
Theory and Applications of Categories
It is these partial dualities that we wish to extend. Second, all are symmetric closed
monoidal categories. All but one are categories of models of a commutative theory and
get their closed monoidal structure from that (see 3.7 below). The theory of Banach balls
is really different from first six and is treated in detail in [Barr, Kleisli, to appear].
What we do here is provide a uniform treatment of the first six examples. We show that
in each case, there is a ∗-autonomous category of uniform space models of the theory. In
most cases, this is equivalent to the topological space models. The main tool used here is
the so-called Chu construction as described in an appendix to the 1979 monograph, [Chu,
1979]. He described in detail a very general construction of a large class of ∗-autonomous
categories. He starts with any symmetric monoidal category V and any object ⊥ therein
chosen as dualizing object to produce a ∗-autonomous category denoted Chu(V, ⊥). The
simplicity and generality of this construction made it appear at the time unlikely that it
could have any real interest beyond its original purpose, namely showing that there was
a plenitude of ∗-autonomous categories. We describe this construction in Section 2.
A preliminary attempt to carry out this approach using the Chu construction appeared
in [Barr, 1996], limited to only two of the seven example categories (vector spaces over
a discrete field and abelian groups). The arguments there were still very ad hoc and
depended on detailed properties of the two categories in question. In this article, we
prove a very general theorem that appeals to very few special properties of the examples.
In 1987 I discovered that models of Jean-Yves Girard’s linear logic were ∗-autonomous
categories. Within a few years, Vaughan Pratt and his students had found out about the
Chu construction and were studying its properties intensively ([Pratt, 1993a,b, 1995,
Gupta, 1994]). One thing that especially struck me was Pratt’s elegant, but essentially
obvious, observation that the category of topological spaces can be embedded fully into
Chu(Set, 2) (see 2.2). The real significance—at least to me—of this observation is that
putting a Chu structure on a set can be viewed as a kind of generalized topology.
A reader who is not familiar with the Chu construction is advised at this point to read
Section 2. Thinking of a Chu structure as a generalized topology leads to an interesting
idea which I will illustrate in the case of topological abelian groups. If T is an abelian group
(or, for that matter, a set), a topology is given by a collection of functions from the point
set of T to the Sierpinski space—the space with two points, one open and the other not.
From a categorical point of view, might it not make more sense to replace the functions to a
set by group homomorphisms to a standard topological group, thus creating a definition
of topological group that was truly intrinsic to the category of groups. If, for abelian
groups, we take this “standard group” to be the circle group R/Z, the resultant category
is (for separated groups) a certain full subcategory of Chu(Ab, R/Z) called chu(Ab, R/Z).
This category is not the category of topological abelian groups. Nonetheless the category
of topological abelian groups has an obvious functor into chu(Ab, R/Z) and this functor
has both a left and a right adjoint, each of which is full and faithful. Thus the category
chu(Ab/, R/Z) is equivalent to two distinct two full subcategories of abelian groups, each
of which is thereby ∗-autonomous. In fact, any topological abelian group that can be
embedded algebraically and topologically into a product of locally compact groups has
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both a finest and a coarsest topology that induce the same set of characters. The two
subcategories consist of all those that have the finest topology and those that have the
coarsest. These are the images of the left and right adjoint, respectively.
1.1. Acknowledgment. I would like to thank Heinrich Kleisli with whom I had a many
helpful and informative discussions on many aspects of this work during one month visits
to the Université de Fribourg during the springs of 1997 and 1998. In particular, the
correct proof of the existence of the Mackey uniformity (or topology) was worked out
there in the context of the category of balls, see [Barr, Kleisli, 1999].
2. The Chu construction
There are many references to the Chu construction, going back to [Chu, 1979], but see
also [Barr, 1991], for example. In order to make this paper self-contained, we will give
a brief description here. We stick to the symmetric version, although there are also
non-symmetric variations.
2.1. The category Chu(V, ⊥). Suppose that V is a symmetric closed monoidal category
and ⊥ is a fixed object of V. An object of Chu(V, ⊥) is a pair (V, V 0 ) of objects of V
together with a homomorphism, called a pairing, h−, −i: V ⊗ V 0 −→ ⊥. A morphism
(f, f 0 ): (V, V 0 ) −→(W, W 0 ) consists of a pair of arrows f : V −→ W and f 0 : W 0 −→ V 0 in V
that satisfy the symbolic identity hf v, w0 i = hv, f 0 w0 i. Diagrammatically, this can be
expressed as the commutativity of the diagram
V ⊗W
0
0
V ⊗ fV ⊗V0
f ⊗ W0
?
W ⊗ W0
h−, −i
h−, −i
?
-⊥
Using the transposes V −→ V 0 −◦ ⊥ and V 0 −→ V −◦ ⊥ of the structure maps, this condition
can be expressed as the commutativity of either of the squares
f
V
?
V 0 −◦ ⊥
-W
?
- W 0 −◦ ⊥
f −◦ ⊥
0
W0
?
W −◦ ⊥
f0
- V0
?
- V −◦ ⊥
f −◦ ⊥
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Theory and Applications of Categories
A final formulation of the compatibility condition is that
Hom((V, V 0 ), (W, W 0 ))
- Hom(V, W )
?
Hom(W 0 , V 0 )
?
- Hom(V ⊗ W 0 , ⊥)
is a pullback.
The internal hom is gotten by using an internalization of the last formulation. Define
the object [(V, V 0 ), (W, W 0 )] of V as a pullback
[(V, V 0 ), (W, W 0 )]
- V −◦ W
?
W 0 −◦ V 0
?
- V ⊗ W 0 −◦ ⊥
and then define
(V, V 0 ) −◦(W, W 0 ) = ([(V, V 0 ), (W, W 0 )], V ⊗ W 0 )
Since the dual of (V, V 0 ) is (V 0 , V ), it follows from the interdefinability of tensor and
internal hom in a ∗-autonomous category that the tensor product is
(V, V 0 ) ⊗ (W, W 0 ) = (V ⊗ W, [(V, V 0 ), (W 0 , W )])
The result is a ∗-autonomous category. See [Barr, 1991] for details.
2.2. The category Chu(Set, 2). An object of Chu(Set, 2) is a pair (S, S 0 ) together with
a function S × S 0 −→ 2. This is equivalent to a function S 0 −→ 2S . When this function
is injective we say that (S, S 0 ) is extensional and then S 0 is, up to isomorphism, a set of
subsets of S. Moreover, one easily sees that if (S, S 0 ) and (T, T 0 ) are both extensional,
then a function f : S −→ T is the first component of some (f, f 0 ): (S, S 0 ) −→(T, T 0 ) if and
only if U ∈ T 0 implies f −1 (U ) ∈ S 0 and then f 0 = f −1 is uniquely determined. This
explains Pratt’s full embedding of topological spaces into Chu(Set, 2).
2.3. The category chu(V, ⊥). Suppose V is a closed symmetric monoidal category as
above and suppose there is a factorization system E/M on V. (See [Barr, 1998] for a
primer on factorization systems.) In general we suppose that the arrows in E are epimorphisms and that those in M are compatible with the internal hom in the sense that if V
−→ V 0 belongs to M, then for any object W , the induced W −◦ V −→ W −◦ V 0 also belongs to M. In all the examples here, E consists of the surjections (regular epimorphisms)
and M of the injections (monomorphisms), for which these conditions are automatic. An
object (V, V 0 ) of the Chu category is said to be M-separated, or simply separated, if the
transpose V −→ V 0 −◦ ⊥ is in M and M-extensional, or simply extensional, if the other
transpose V 0 −→ V −◦ ⊥ is in M. We denote by Chus (V, ⊥), Chue (V, ⊥), and Chuse (V, ⊥)
5
the full subcategories of separated, extensional, and separated and extensional, respectively, objects of Chu(V, ⊥). Following Pratt, we usually denote the last of these by
chu(V, ⊥).
The relevant facts are
1. The full subcategory Chus (V, ⊥) of separated objects is a reflective subcategory of
Chu(V, ⊥) with reflector s.
2. The full subcategory Chue (V, ⊥) of extensional objects is a coreflective subcategory
of Chu(V, ⊥) with coreflector e.
3. If (V, V 0 ) is separated, so is e(V, V 0 ); if (V, V 0 ) is extensional, so is s(V, V 0 ).
4. Therefore chu(V, ⊥) is both a reflective subcategory of the extensional category and
a coreflective subcategory of the separated subcategory. It is, in particular, complete
and cocomplete.
5. The tensor product of extensional objects is extensional and the internal hom of an
extensional object into a separated object is separated.
6. Therefore by using s(− ⊗ −) as tensor product and r(− −◦ −) as internal hom, the
category chu(V, ⊥) is ∗-autonomous.
For details, see [Barr, 1998].
3. Topological and uniform space objects
3.1. Topology and duality. In a ∗-autonomous category we have, for any object A,
that A∗ ∼
= > −◦ A∗ ∼
= A −◦ >∗ . If we denote >∗ by ⊥, we see that the duality has the form
A 7→ A −◦ ⊥. The object ⊥ is called the dualizing object and, as we will see, the way
(or at least a way) of creating a duality is by finding a dualizing object in some closed
monoidal category.
In order that a category have a duality realized by an internal hom, there has to be a
way of constraining the maps so that the dual of a product is a sum. In an additive category, for example, this happens without constraint for finite products, but not normally
for infinite ones. A natural constraint is topological. If, for example, the dualizing object
is finite discrete, then any continuous map from a product can depend on only finite many
of the coordinates.
For example, even for ordinary
Q
Q topological spaces, for a continuous
−1
function f : Xi −→ 2, f (0) has the form
/ Xi where J is a finite
Q Y × i∈J
Q subset of I
−1
and Y Q
is a subset
of
the
finite
product
X
.
But
then
f
(1)
=
Z
×
i
i∈J
i∈J
/ Xi , where
0
0
Z =
i∈J Xi − Y . If two elements of the product x = (xi )i∈I and x = (xi )i∈I are
0
0
elements such that xi = xi for i ∈ J then either x ∈ Y and x ∈ Y or x ∈ Z and x0 ∈ Z,
but in either case f x = f x0 . Thus f depends Q
only the coordinates belonging to J, which
means f factors through the finite product i∈J Xi . A similar argument works if 2 is
replaced by any finite discrete set.
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Theory and Applications of Categories
3.2. Uniformity. Despite the examples above there are reasons for thinking that the
technically “correct” approach to duality is through the use of uniform structures. A
very readable and informative account of uniform spaces is in [Isbell, 1964]. However,
Isbell uses uniform covers as his main definition. Normally I prefer uniform covers to the
approach using entourages, but for our purposes here entourages are more appropriate.
For any equational category, a uniform space object is an object of the category
equipped with a uniformity for which the operations of the theory are uniform. A morphism of uniform space objects is a uniform function that is also a morphism of models
of the theory. Topological space objects are defined similarly. Any uniformity leads to
a topological space object in a canonical way and uniform functions become continuous
functions for that canonical topology. But not every topology comes from a uniformity
and, if it does, it is not necessarily from a unique one. For example, the metric on the
space of integers and on the set of reciprocals of integers both give the discrete topology, but the associated uniformities are quite different. (Metric spaces have a canonical
uniformity. See Isbell’s book for details.)
If, however, there is an abelian group structure among the operations of an equational
theory, there is a canonical uniformity associated to every topology. Namely, for each
open neighborhood U of the group identity take {(x, y) | xy −1 ∈ U } as an entourage.
Moreover, a homomorphism of the algebraic structure between uniform space objects is
continuous if and only if it is uniform. Thus there is no difference, in such cases, between
categories of uniform and topological space objects. Obviously, the category of topological
space objects is more familiar. However, one of our categories, semilattices, does not have
an abelian group structure and for that reason, we have cast our main theorem in terms
of uniform structure. There is another, less important, reason. At one point, in dealing
with topological abelian groups, it becomes important that the circle group is complete
and completeness is a uniform, not topological, notion.
If V is an equational category, we denote by Un(V), the category of uniform objects
of V. We let | − |: Un(V) −→ V to be functor that forgets the uniform structure.
3.3. Small entourage. Let A be a uniform V object. An entourage E ⊆ A × A is
called a small entourage if it contains no subobject of A × A that properly contains the
diagonal and if any homomorphism f : B −→ A of uniform V objects for which (f ×f )−1 (E)
is an entourage of B is uniform.
3.4.
. In all the examples we will be considering, there will be a given class of uniform
objects D and we will be dealing with the full subcategory A of Un(V) consisting of those
objects strongly cogenerated by D, which is to say that those that can algebraically
and uniformly embedded into a product of objects of D.
3.5. Half-additive categories. A category is called half-additive if its hom functor
factors through the category of commutative monoids. It is well known that in any such
category finite sums are also products (see, for example, [Freyd, Scedrov, 1990], 1.59). Of
course, additive categories are half-additive. Of the six categories considered here, five
are additive and one, semilattices, is not additive, but is still half-additive. In fact, a
7
semilattice is a commutative monoid in which every element is idempotent. This monoid
structure can be equally well viewed as sup or inf.
3.6. The closed monoidal structure. The categories we are dealing with are all symmetric closed monoidal. With one exception, the closed structure derives from their being
models of a commutative theory.
3.7. Commutative theories. A commutative theory is an equational theory whose
operations are homomorphisms ([Linton, 1966]). Thus in any abelian group G, as contrasted with a non-abelian group, the multiplication G × G −→ G is an abelian group
homomorphism, as are all the other operations.
3.8. Theorem. [Linton] The category of models of a commutative theory has a canonical
structure of a symmetric closed monoidal category.
Proof. Suppose V is the category of models and U : V −→ Set is the underlying set
functor with left adjoint F . If V and W are objects of V, then W −◦ V is a subset of V U W
defined as the simultaneous equalizer, taken over all operations ω of the theory, of
V UW
@
- (V n )(U W )n
@
V ω@
@
n
ω (U W )
@
@
R ?
n
V (U W )
Here n is the arity of ω and the top arrow is raising to the nth power. Since the theory is
commutative, ω is a homomorphism and so the equalizer is a limit of a diagram in V and
hence lies in V. In particular, the internal hom of two objects of V certainly lies in V. The
free object on one generator is the unit for this internal hom. As for the tensor product,
V ⊗ W is constructed as a quotient of F (U V × U W ), similar to the usual construction of
the tensor product of two abelian groups.
3.9. Proposition. If A and B are objects of an equational category V equipped with
uniformities for which their operations are uniform, then the set of uniform morphisms
from A −→ B is a subobject of |A| −◦ |B| and thus the category of uniform V objects is
enriched over V. It also has tensors and cotensors from V.
Proof. Let [A, B] denote the set of uniform homomorphisms from A to B. For each nary operation ω, the arrow ωB: B n −→ B is a uniform homomorphism and hence an arrow
[A, B]n ∼
= [A, B n ] −→[A, B] is induced by ωB and we define this as ω[A, B]. This presents
[A, B] as a subobject of |A| −◦ |B| so that it also satisfies all the equations of the theory
and is thus an object of V. The cotensor AV is given by the object V −◦ |A| equipped
with the uniformity induced by AU V . The tensor is constructed using the adjoint functor
theorem with all uniformities on V ⊗ |A| as solution set.
8
Theory and Applications of Categories
4. The main theorem
We are now ready to state our main theorem.
4.1. Theorem. Suppose V is an equational category equipped with a closed monoidal
structure, D is a class of uniform space objects of V and A is the full subcategory of the
category of uniform space objects of V that is strongly cogenerated by D. Suppose that ⊥
is an object of A with the following properties
1. V is half-additive.
2. D is closed under finite products.
3. ⊥ has a small entourage.
4. The natural map > −→[⊥, ⊥] is an isomorphism.
5. If D is an object of D, A ⊆ D is a subobject, then the induced arrow [D, ⊥] −→[A, ⊥]
is surjective.
6. For every object D of D, the natural evaluation map D −→ ⊥[D,⊥] is injective.
7. A is enriched over V and has cotensors from V.
Then, using the regular-epic/monic factorization system, the canonical functor P : A
−→ chu(V, |⊥|) defined by P (A) = (|A|, [A, ⊥]) has a right adjoint R and a left adjoint L,
each of which is full and faithful.
4.2.
. Before beginning the proof, we make some observations. We will call a morphism A
−→ ⊥ a functional on A. In light of condition 5, condition 6 need be verified only for
objects that are algebraically 2-generated (and in the additive case, only for those that
are 1-generated) since any separating functional can be extended to all of D.
In all our examples, ⊥ is complete and a closed subobject of an object of D belongs
to D so that it is sufficient to verify condition 5 when A belongs to D.
The conclusion of the theorem implies that the full images of both R and L are
equivalent to chu(V, ⊥) and hence both image categories are ∗-autonomous.
The diagonal of A in A × A will be denoted ∆A . We begin the proof with a lemma.
Q
4.3. Lemma. Suppose that A ⊆ i∈I Ai and ϕ: A −→ ⊥ is a uniform functional. Then
e is the image of A −→ Q Ai −→ Q Ai with
there is a finite subset J ⊆ I such that if A
i∈I
i∈J
ϕ
e
e −−→ ⊥.
the subspace uniformity, then ϕ factors as A −→ A
9
Proof. Let E ⊆ ⊥ × ⊥ be a small entourage. The definition Q
of the product uniformity
Q
implies that there is a finite subset J ⊆ I such that if we let B = i∈J Ai and C = i∈J
/ Ai ,
−1
then there is an entourage F ⊆ B × B for which (A × A) ∩ (F × (C × C)) ⊆ (ϕ × ϕ) (E).
But then (A×A)∩(∆B ×(C ×C)) is a subobject of A×A that is included in (ϕ×ϕ)−1 (E).
This implies that (ϕ × ϕ)((A × A) ∩ (∆B × (C × C))) is a subobject of ⊥ × ⊥ lying between
∆⊥ and E, which is then ∆⊥ . In particular, if a = (ai ) and a0 = (a0i ) are two elements of A
such that ai = a0i for all i ∈ J, then ϕ(a) = ϕ(a0 ). Thus, ignoring the uniform structure, we
e −→ ⊥. But (A
e × A)
e ∩ F ⊆ (ϕ
can factor ϕ via an algebraic homomorphism ϕ:
e A
e × ϕ)
e −1 (E)
e
which means that ϕ
e is uniform in the induced uniformity on A.
4.4. Corollary. For any A ⊆ B in A, the induced [B, ⊥] −→[A, ⊥] is surjective.
Q
Proof. Since there is an embedding
B
⊆
i∈I Di with Di objects of D, it is sufficient
Q
to do this in the case that B = i∈I Di . The lemma says that any functional in [A, ⊥]
e −→ ⊥ where, for some finite J ⊆ I, A
e ⊆ Q Di . The latter object is
factors as A −→ A
i∈J
in D by condition 2 and the map extends to it by condition 5.
P
4.5.QCorollary. For any set {Ai | i ∈ I} of objects of A, the canonical map i∈I [Ai , ⊥]
−→[ i∈I Ai , ⊥] is an isomorphism.
Q
Proof. By taking A = i∈I Ai in the lemma, we see that every functional
Q on the
product
factors
through
a
finite
product.
That
is,
the
canonical
map
colim
[
J⊆I
i∈J Ai , ⊥]
Q
−→[ i∈I Ai , ⊥] is an isomorphism, where the colimit is taken over the finite subsets J ⊆ I.
On the other hand, half-additivity implies that the canonical map from a finite sum to
finite product is an isomorphism. Putting these together, we conclude that
"
#
X
X
Y
X
[Ai , ⊥] ∼
[Ai , ⊥] ∼
[Ai , ⊥] ∼
Ai , ⊥
= colimJ⊆I
= colimJ⊆I
= colimJ⊆I
i∈I
i∈J
∼
= colimJ⊆I
"
Y
i∈J
i∈J
#
Ai , ⊥ ∼
=
"
Y
i∈J
#
Ai , ⊥
i∈I
4.6. Proof of the theorem. The right adjoint to P is defined as follows. If (V, V 0 )
is an object of chu(V, ⊥), then by definition of separated V −→ V −◦ V 0 is monic. The
underlying functor from the category of uniform objects to V has a left adjoint and hence
0
preserves monics so that |V | −→[V 0 , ⊥] is also monic. Since the latter is a subset of |⊥||V | ,
0
we have that |V | ⊆ |⊥|V | |. Then we let R(V, V 0 ) denote |V |, equipped with the uniformity
0
induced as a uniform subspace of ⊥|V | . Also denote by σ(V, V 0 ) the uniformity of R(V, V 0 ).
This is the coarsest uniformity on V for which all the functionals in V 0 are uniform. A
morphism P A −→(V, V 0 ) consists of an arrow f : |A| −→ V in V such that for any ϕ ∈ V 0
0
the composite ϕøf is uniform. This means that the composite A −→ V −→ ⊥U V is uniform
and hence that A −→ R(V, V 0 ) is. Conversely, if f : A −→ R(V, V 0 ) is given, then we have
0
f : |A| −→ V such that the composite A −→ V −→ ⊥U V is uniform and if we follow it by the
10
Theory and Applications of Categories
coordinate projection corresponding to ϕ ∈ V 0 , we get that ϕøf is uniform for all ϕ ∈ V 0 ,
so that there is induced a unique arrow V 0 −→[A, ⊥] as required. This shows that R is
right adjoint to P .
Next we claim that P R is naturally equivalent to the identity. This is equivalent to
showing any functional uniform on R(V, V 0 ) already belongs to V 0 . But any functional
0
ϕ: R(V, V 0 ) −→ ⊥ extends by Corollary 4.4 to a functional on ⊥V . From Corollary 4.5,
there is a finite set of functionals ϕ1 , . . . , ϕn ∈ V 0 and a functional α: ⊥n −→ ⊥ such that
- ⊥V 0
A
ϕ
?
⊥ α
?
⊥n
where the right hand arrow is the projection on the coordinates corresponding to ϕ1 , . . . ,
ϕn . If the components of α are α1 , . . . , αn , then this says that ϕ = α1 øϕ1 + · · · + αn øϕn ,
which is in V 0 .
For an object A of A it will be convenient to denote RP A by σA. This is the underlying
V object of A equipped with the weak uniformity for the functionals on A.
Define a homomorphism f : A −→ B to be weakly uniform if the composite A −→ B
−→ σB is uniform. This is equivalent to the assumption that for every functional ϕ: B
f
ϕ
−→ ⊥, the composite A −−→ B −−→ ⊥ is a functional on A. It is also equivalent to the
assumption that f : σA −→ σB is uniform. Given A, let {A −→ Ai | i ∈ I} range over the
isomorphism classes of weakly uniform surjective homomorphisms out of A. Define τ A as
the pullback in the diagram
Q
- Ai
τA
?
σA
-
Q?
σAi
If f : A −→ B is weakly uniform, it factors A −
→
→A0 ⊆ B and the first arrow is weakly
uniform since every uniform functional on A0 extends to a uniform functional on B. Since,
up to isomorphism, A −
→
→A0 is among the A −
→
→Ai , it follows that f : τ A −→ B is uniform.
Since the identity is a weakly uniform surjection, the lower arrow in the square above is
a subspace inclusion and hence so is the upper arrow. That implies that the lower arrow
in the diagram of functionals
Q
- [σA, ⊥]
[ σAi , ⊥]
Q ?
[ Ai , ⊥]
?
- [τ A, ⊥]
11
P
P
is a surjection. The left hand arrow is equivalent to
[σAi , ⊥] −→ [Ai , ⊥] (Corollary 4.5), which we have just seen is an isomorphism. Thus the right hand arrow is a
surjection, while it is evidently an injection. This shows that τ A has the same functionb were a strictly finer uniformity than that of τ A on the same underlying
als as A. If A
b would
V structure that had the same set of functionals as A, then the identity A −→ A
b would be uniform, a contradiction. Thus if we
be weakly uniform and then τ A −→ A
0
0
define L(V, V ) = τ R(V, V ) we know at least that P L ∼
= Id so that L is full and faith0
0
ful. If (f, f ): (V, V ) −→ P A is a Chu morphism, then f : V −→ |A| is a homomorphism
such that for each uniform functional ϕ: A −→ ⊥ the composite ϕøf ∈ V 0 . Thus R(V, V 0 )
−→ A is weakly uniform and hence L(V, V 0 ) = τ R(V, V 0 ) −→ A is uniform. Conversely,
if f : L(V, V 0 ) −→ A is uniform, then for any functional ϕ: A −→ ⊥, the composite ϕøf is
uniform on L(V, V 0 ) and hence belongs to V 0 so that we have (V, V 0 ) −→ P A.
We will say that an object A with A = σA has the weak uniformity (or weak
topology) and that one for which A = τ A has the Mackey uniformity (or Mackey
topology). The latter name is taken from the theory of locally convex topological vector
spaces where a Mackey topology is characterized by the property of having the finest
topology with a given set of functionals.
4.7. Exceptions. In verifying the hypotheses of Theorem 4.1, one notes that each example satisfies simpler hypotheses. And each simpler hypothesis is satisfied by most of
the examples. Most of the categories are additive (exception: semilattices) and then we
can use topologies instead of uniformities. In most cases, the dualizing object is discrete
(exceptions: abelian groups and real or complex vector spaces) and the existence of a
small entourage (or neighborhood of 0 in the additive case) is automatic. In most cases,
the theory is commutative and the closed monoidal structure comes from that (exception:
modules over a Hopf algebra) so that the enrichment of the uniform category over the
base is automatic. Thus most of the examples are exceptional in some way (exceptions:
vector spaces over a field and modules with a dualizing module), so that we may conclude
that they are all exceptional.
5. Vector spaces: the case of a discrete field
The simplest example of the theory is that of vector spaces over a discrete field. Let K be
a fixed discrete field. We let V be the category of K-vector spaces with the usual closed
monoidal structure and let D be the discrete spaces. Since the category is additive, we
can work with topologies rather than uniformities. We take the dualizing object ⊥ as the
field K with the discrete topology.
The conditions of Theorem 4.1 are all evident and so we conclude that the full subcategories of the category of topological K-vector spaces consisting of weakly topologized
space and of Mackey spaces are ∗-autonomous.
We note that infinite dimensional discrete spaces do not have the weak uniformity. In
fact the weak uniformity associated to the chu space (V, V −◦ K) is V with the uniform
12
Theory and Applications of Categories
topology in which the open subspaces are the cofinite dimensional ones. Since the 0
subspace is not cofinite dimensional, the space is not discrete. On the other hand, the
map to the discrete V is weakly uniform and so the Mackey space associated is discrete.
6. Dualizing modules
The case of a vector space over a discrete field has one generalization, suggested by R.
Raphael (private communication). Let R be a commutative ring. Say that an R-module ⊥
is a dualizing module if it is a finitely generated injective cogenerator and the canonical
map R −→ HomR (⊥, ⊥) is an isomorphism. Let A be the category of topological (=
uniform) R-modules that are strongly cogenerated by the discrete ones. Then taking
the small neighborhood to be {0} and D the class of discrete modules, the conditions of
Theorem 4.1 are satisfied and we draw the same conclusion.
6.1. Existence of dualizing modules. Not every ring has a dualizing module. For
example, no finitely generated abelian group is injective as a Z-module, so Z lacks a
dualizing module. On the other hand, If R is a finite dimensional commutative algebra
over a field K, then HomK (R, K) is a dualizing module for K. It follows that any artinian
commutative ring has a dualizing module:
6.2. Proposition. Suppose K is a commutative ring with a dualizing module D and R
is a commutative K-algebra finitely generated and projective as a K-module. Then for
any finitely generated R-projective P of constant rank one, the R-module HomK (P, D) is
a dualizing module for R.
Proof. It is standard that such a module is injective. In fact, for an injective homomorphism f : M −→ N , we have that
HomR (f, HomK (P, D)) ∼
= HomK (P ⊗R f, D)
which is surjective since P is R-flat. Since P is finitely generated projective as an Rmodule, it is retract of a finite sum of copies of R. Similarly, R is a retract of a finite sum
of copies of K, whence P is a retract of a finite sum of copies of K. Then Hom(P, D) is
a retract of a finite sum of copies of D and is thus finitely generated as a K-module, a
fortiori as an R-module. Next we note that a constantly rank one projective P has endomorphism ring R. In fact, the canonical R −→ HomR (P, P ) localizes to the isomorphism
RQ −→ HomRQ (PQ , PQ ) ∼
= HomRQ (RQ , RQ ) which is an isomorphism, at each prime ideal
Q and hence is an isomorphism. Then we have that
HomR (HomK (P, D), HomK (P, D)) ∼
= HomK (P ⊗R HomK (P, D), D)
∼
= HomR (P, HomK (HomK (P, D), D))
∼
= HomR (P, P ) ∼
=R
since D is a dualizing module for K and P is a finitely generated K module.
13
Whether any non-artinian commutative ring has a dualizing module is an open question. For example, the product of countably many fields does not appear to have a
dualizing module. The obvious choice would be the product ring itself and, although it
is injective, it is not a cogenerator since the quotient of the ring mod the ideal which is
the direct sum is a module that is annihilated by every minimal idempotent so that the
quotient module has no non-zero homomorphism into the ring.
7. Vector spaces: case of the real or complex field
We will treat the case of the complex field. The real case is similar. We take for D the class
of Banach spaces and the base field C as dualizing object. The D-cogenerated objects are
just the spaces whose topology is given by seminorms. These are just the locally convex
spaces (see [Kelly, Namioka, 1963], 2.6.4). The conditions of 4.1 follow immediately from
the Hahn-Banach theorem and we conclude that the category chu(V, ⊥) is equivalent to
both full subcategories of weakly topologized and Mackey spaces and that both categories
are ∗-autonomous. In particular, the existence of the Mackey topology follows quite easily
from this point of view.
We can also give a relatively easy proof of the fact that the Mackey topology is
convergence on weakly compact, convex, circled subsets of the dual. In fact, let A be a
locally convex space and A∗ denote the weak dual. If f : A −→ D is a weakly continuous
map, then we have an induced map, evidently continuous in the weak topology, f ∗ : D∗
−→ A∗ and one sees immediately that the weakly continuous seminorm induced on A
|| − ||
f
by the composite A −−→ D −−−−−→ R is simply the sup on f ∗ (C), where C is the unit
ball of D∗ , which is compact in the weak topology. On the other hand, if C ⊆ A∗ is
compact, convex, and circled, let B be the linear subspace of A∗ generated by C made
into a Banach space with C as unit ball. With the topology induced by that of C, so
that a morphism out of B is continuous if and only if its restriction to C is, B becomes
an object of the category C as described in [Barr, 1979], IV.3.10. This category consists
of the mixed topology spaces whose unit balls are compact. The discussion in IV.3.16 of
the same reference then implies that every functional on B ∗ is represented by an element
of B. This means that the induced A −→ B ∗ is weakly continuous. But B ∗ is a Banach
space whose norm is the absolute sup on C, as is the induced seminorm on A.
8. Banach balls
A Banach ball is the unit ball of a Banach space. The conclusions, but not the hypotheses
of Theorem 4.1 are valid in this case too. However, the proof is different and will appear
elsewhere [Barr, Kleisli, 1999]. The proof given here of the existence of the right adjoint
and the Mackey topology was first found in this context.
14
Theory and Applications of Categories
9. Abelian groups
The category of abelian groups is an example of the theory. For D we take the class of
locally compact groups. The dualizing object is, as usual, the circle group, R/Z. Since
the category is additive, we can deal with topologies instead of uniformities. A small
neighborhood of 0 is an open neighborhood of 0 that contains no non-zero subgroup
and for which a homomorphism to the circle is continuous if and only if the inverse image
of that neighborhood is continuous.
9.1. Proposition. The image U ⊆ R/Z of the interval (−1/3, 1/3) ⊆ R is a small
neighborhood of 0.
Proof. Suppose x 6= 0 in U . Suppose, say, that x is in the image of a point in (0, 1/3).
Then the first one of x, 2x, 4x, . . . , that is larger than 1/3 will be less than 2/3 ≡ −1/3,
which shows that U contains no non-zero subgroup.
It is clear that the set of all 2−n U, n = 0, 1, 2, . . . is a neighborhood base at 0. Suppose
that f : A −→ R/Z is a homomorphism such that V = f −1 (U ) is open in A. Let V0 = V
and inductively choose an open neighborhood Vn of 0 so that Vn − Vn ⊆ Vn−1 . Then one
easily sees by induction that Vn ⊆ f −1 (2−n U ).
The remaining conditions of Theorem 4.1 are almost trivial, given Pontrjagin duality.
The only thing of note is that if D ∈ D and A ⊆ D then any continuous homomorphism
ϕ: A −→ R/Z can first be extended to the closure of A, since the circle is compact and
hence complete in the uniformity. A closed subgroup of a locally compact group is locally
compact and the duality theory of locally compact groups gives the extension to all of D.
9.2. Proposition. Locally compact groups are Mackey groups.
Proof. Since all the groups in A are embedded in a product of locally compact groups,
it suffices to know that a weakly continuous map between locally compact topological
groups is continuous. This is found in [Glicksberg, 1962].
9.3. Other choices for D. One thing to note is that it is possible to choose a different
category A. The result can be a different notion of Mackey group. For instance, you could
choose for A the subspaces of compact spaces. In that case weakly continuous coincides
with continuous and weak and Mackey topologies coincide. Another possibility is to use
compact and discrete spaces. It is easy to see that the real line cannot be embedded
into a product of compact and discrete spaces. There are no non-zero maps to a discrete
space, so it would have to embedded into a product of compact spaces. But the real line is
complete, so the only way it could be embedded into a product of compact spaces would
be if it were compact.
In the original monograph, countable sums of copies of R were permitted in D. But
the sum of countably many copies of R is not locally compact. Here we show that we also
get a model of the theory by allowing D to consist of countable sums of locally compact
groups. The only issue here is the injectivity of the circle. So suppose A ⊆ D, where
D = D1 ⊕D2 ⊕· · · is a countable sum of locally compact groups. As above, we can suppose
15
that A is closed in D. Let Fn (D) = D1 ⊕ · · · ⊕ Dn and Fn (A) = A ∩ Fn (D). Every element
of A is in some finite summand, so that, algebraically at least, A = colim Fn (A). Whether
it is topologically is not important, since we will show that every continuous character
on the colimit extends to a continuous character on D. What does matter is that, by
definition of the topology on the countable sum, D = colim Fn (D) both algebraically and
topologically. The square
- Fn (A)
Fn−1 (A)?
?
?
Fn−1 (D)-
?
- Fn (D)
is a pullback by definition. There is no reason for it to be a pushout, but if we denote
the pushout by Pn , it is trivial diagram chase to see that Pn )−→ Fn (D) is injective. The
group Fn (D) is locally compact and so, therefore, is the closed subgroup Fn (A) and so is
the closure Pn . Thus, taking Pontrjagin duals, all the arrows in the diagram below are
surjective and the square is a pullback:
Fn (D)∗
A@HH
A @ HH
H
A @
HH
A @
HH
R
@
j
H
R
j
A @
∗
P
Fn−1 (D)∗
n
A
A
A
A
UA ?
A
?
?
U?
A
∗ Fn (A)
Fn−1 (A)∗
The surjectivity of the arrow Fn (D)∗ −
→
→Pn ∗ implies that each square of
···
- Fn+1 (D)∗ - Fn (D)∗ - Fn−1 (D)∗
- ···
?
?
···
?
?
?
?
?
?
- Fn+1 (A)∗ - Fn (A)∗ - Fn−1 (A)∗
?
?
- ···
is a weak pullback. From this, it is a simple argument to see that the induced arrow
lim Fn (D) −
→
→lim Fn (A) is surjective.
10. Modules over a cocommutative Hopf algebra
Modules over a Hopf algebra are not models of a commutative theory, unless the algebra
should also be commutative and, even so, the closed structure does not come from there.
16
Theory and Applications of Categories
Thus we will have to verify directly that the category of the topological algebras is enriched
over the category of discrete algebras for the theory.
There are two important special cases and we begin with brief descriptions of them.
10.1. Group representations. Let G be a group and K be a field. A K-representation
of G is a homomorphism of G into the group of automorphisms of some vector space over
K. Equivalently, it is the action of G on a K-vector space. A third equivalence is with
a module over the group algebra K[G]. The category of K-representations of G is thus
an equational category, that of the K[G]-modules, but the theory is not commutative
unless G should be commutative and even in that case, we do not want that closed
monoidal structure. The one we want has as tensor product of modules M and N the
K-tensor product M ⊗ N = M ⊗K N . The G-action is the so-called diagonal action
x(m ⊗ n) = xm ⊗ xn, x ∈ G, extended linearly. The internal hom takes for M −◦ N the
set of K-linear maps with action given by (xf )m = x(f (x−1 m)) for x ∈ G. This gives a
symmetric closed monoidal structure for which the unit object is K with trivial G action,
meaning every element of G acts as the identity on K.
If M and N are topological vector spaces with continuous action of G (which is assumed discrete, at least here), then it is easily seen that the set of continuous linear
transformations M −→ N is a G-representation with the same definition of action and we
denote it by [M, N ] as before. Thus the category of topological (= uniform) G-modules is
enriched over the category of G-modules. The cotensor is also easy. Define AV as |A| −◦ V
topologized as a subspace of AU V .
10.2. Lie algebras. Let K be a field and g be a K-Lie algebra. A K-representation of g
is a Lie algebra homomorphism of g into the Lie algebra of endomorphisms of a K-vector
space V . In other words, for x ∈ g and v ∈ V , there is defined a K-linear product xv
in such a way that [x, y]v = x(yv) − y(xv). If V and W are two such actions, there is
an action on V ⊗ W = V ⊗K W given by x(v ⊗ w) = xv ⊗ w + v ⊗ xw. The space
V −◦ W of K-linear transformations has an action given by (xf )(v) = x(f v) − f (xv). If
g acts continuously on topological vector spaces V and W , then xf is continuous when
f is so that the category of topological representations is enriched over the category of
representations. The cotensor works in the same way as with the groups.
10.3. Modules over a cocommutative Hopf algebra. These two notions above come
together in the notion of a module over a cocommutative Hopf algebra. Let K be a field. A
cocommutative Hopf algebra over K is a K-algebra given by a multiplication µ: H ⊗H
−→ H (all tensor products in this section are over K), a unit η: K −→ H, a comultiplication
δ: H −→ H ⊗ H, a counit : H −→ K and an involution ι: H −→ H such that
HA–1. (H, µ, η) is an associative, unitary algebra;
HA–2. (H, δ, ) is a coassociative, counitary, cocommutative coalgebra;
HA–3. δ and are algebra homomorphisms; equivalently, µ and η are coalgebra homomorphisms;
17
HA–4. ι makes (H, µ, η) into a group object in the category of cocommutative coalgebras.
This last condition is equivalent to the commutativity of
H
δ - H ⊗H
η◦
?
Hµ
1⊗ι
?
H ⊗H
The leading examples of Hopf algebras are group algebras and the enveloping algebras
of Lie algebras. If G is a group, the group algebra K[G] is a Hopf algebra with δ(x) = x⊗x,
(x) = 1 and ι(x) = x−1 , all for x ∈ G and extended linearly. In the case of a Lie algebra
g, the definitions are δ(x) = 1 ⊗ x + x ⊗ 1, (x) = 0 and ι(x) = −x, all for x ∈ g.
10.4. The general case. Let H be a cocommutative Hopf algebra. By an H-module
we simply mean a module over the algebra part of H. If M and N are modules, we define
M ⊗ N to be the tensor product over K with H action given by the composite
δ⊗1⊗1
H ⊗ M ⊗ N −−−−−−−−→ H ⊗ H ⊗ M ⊗ N −→ H ⊗ M ⊗ H ⊗ N −→ M ⊗ N
The second arrow is the symmetry isomorphism of the tensor and the third is simply
the two actions. We define M −◦ N to be the set of K-linear arrows with the action
H ⊗ (M −◦ N ) −→ M −◦ N the transpose of the arrow H ⊗ M ⊗ (M −◦ N ) −→ N given by
δ⊗1⊗1
H ⊗ M ⊗ (M −◦ N ) −−−−−−−−→ H ⊗ H ⊗ M ⊗ (M −◦ N )
1⊗ι⊗1⊗1
−−−−−−−−−−−→ H ⊗ H ⊗ M ⊗ (M −◦ N )
−→ H ⊗ M ⊗ (M −◦ N ) −→ H ⊗ N −→ N
The third arrow is the action of H on M , the fourth is evaluation and the fifth is the
action of H on N .
That this gives an autonomous category can be shown by a long diagram chase. The
tensor unit is the field with the action xa = (x)a.
We have to show that the category of topological modules is enriched over the category
of modules. We can describe the enriched structure as consisting of the continuous linear
maps with the module structure given as before, that is by conjugation. The continuity
of the module structure guarantees that the action preserves continuity. From then on
the argument is the same. The dualizing object is the discrete field K which has a small
neighborhood and the rest of the argument is the same. The cotensor is just as in the
case of group representations.
The class D consists of the discrete objects. The dualizing object is the tensor unit.
Since the internal hom is just that of the vector spaces, the conditions of Theorem 4.1 are
easy.
18
Theory and Applications of Categories
11. Semilattices
By semilattice we will mean inf semilattice, which is a partially ordered set in which
every finite set of elements has an inf. It is obviously sufficient that there be a top element
and that every pair of elements have an inf. The category is obviously equivalent to that
of sup semilattices, since you can turn the one upside down to get the other. The category
is equational having a single constant, 1 (the top element) and a single binary operation ∧
which is unitary (with respect to 1), commutative, associative and idempotent. (In fact,
sup semilattices have exactly the same description—it all depends on how you interpret
the operations.)
Semilattices do not form an additive category, but they are half-additive since they
are commutative monoids. The dualizing object is the 2 element chain with the discrete
uniformity, which evidently has a small entourage. Since it is the tensor unit, condition 4
of Theorem 4.1 is satisfied. For D, we take the class of discrete lattices. We need show
only conditions 5 and 6.
Suppose we have an inclusion L1 ⊆ L2 of discrete semilattices and f : L1 −→ ⊥ is a
semilattice homomorphism. We will show that if x ∈ L2 − L1 , then f can be extended to
the semilattice generated by L1 and x. This semilattice is L1 ∪ (L1 ∧ x). We first define
f x = 1 unless there are elements a, b ∈ L1 such that f a = 1, f b = 0 and a∧x ≤ b in which
case we define f x = 0. Then we define f (a ∧ x) = f a ∧ f x for any a ∈ L1 . The only thing
we have to worry about is if a ∧ x ∈ L1 for some a ∈ L1 . If f a = 0, then f (a ∧ x) ≤ f a
so that f (a ∧ x) = 0 = f a ∧ f x. If f a = 1, then either f (a ∧ x) = 1 or taking b = a ∧ x
we satisfy the condition for defining f x = 0 and then 0 = f (a ∧ x) = f a ∧ f x as required.
The rest of the argument is a routine application of Zorn’s lemma. This completes the
proof of 5. Now 6 follows immediately since given any two elements of a discrete lattice,
they generate a sublattice of at most 4 four elements and it is easy to find a separating
functional on that sublattice.
12. The category of δ-objects
We will very briefly explain why the categories of δ-objects considered in [Barr, 1979]
is also ∗-autonomous. Of course, it is likely more interesting that the larger categories
constructed here are ∗-autonomous, but in the interests of recovering all the results from
the monograph, we include it.
An object T is called ζ-complete if it is injective with respect to dense subobjects of
compact objects. That is, if in any diagram
C0
?
T
-C
19
with C compact and C0 a dense subobject, can be completed by an arrow C −→ T . The
object T is ζ ∗ -complete if T ∗ is ζ-complete. An object is called a δ-object if is ζ-complete,
ζ ∗ -complete and reflexive.
12.1. Theorem. The full subcategories of δ-objects are ∗-autonomous subcategories the
categories of Mackey objects.
Proof. The proof uses one property that we will not verify. Namely that all compact
objects are δ-objects. The duals of the compact objects are complete (in most cases
discrete). For an object T , we define ζT as the intersection of all the ζ-complete subobjects
of the completion of T . The crucial claim is that if T is ζ-complete, so is (ζT ∗ )∗ . In fact,
the adjunction arrow T ∗ −→ ζT ∗ gives an arrow (ζT ∗ )∗ −→ T ∗∗ ∼
= T . Now consider a
diagram
-C
C0
?
(ζT ∗ )∗
-T
Since T is ζ-complete, there is an arrow C −→ T that makes the square commute. This
gives T ∗ −→ C ∗ and, since C ∗ is complete, ζT ∗ −→ C ∗ , and then C ∼
= C ∗∗ −→(ζT ∗ )∗ , as
required. We now invoke Theorem 2.3 of [Barr, 1996] to conclude that the δ-objects form
a ∗-autonomous category.
References
M. Barr (1979), ∗-Autonomous Categories. Lecture Notes in Mathematics 752.
M. Barr (1991), ∗-Autonomous categories and linear logic.
Computer Science, 1, 159–178.
Mathematical Structures
M. Barr (1996), ∗-Autonomous categories, revisited. J. Pure Applied Algebra, 111, 1–20.
M. Barr (1998), The separated extensional Chu category. Theory and Applications of
Categories, 4.6, 137–147.
M. Barr and H. Kleisli (1999), Topological balls. To appear in Cahiers de Topologie et
Géométrie Différentielle Catégorique, 40.
P.-H. Chu (1979), Constructing ∗-autonomous categories. Appendix to [Barr, 1979].
P.J. Freyd, A. Scedrov (1990), Categories, Allegories. North-Holland.
I. Glicksberg (1962), Uniform boundedness for groups. Canadian J. Math. 14, 269-276.
V. Gupta (1994), Chu Spaces: A Model of Concurrency. Ph.D. Thesis, Stanford
University.
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Theory and Applications of Categories
J.L. Kelley, I. Namioka (and coauthors) (1963), Linear Topological Spaces. Van
Nostrand.
F.E.J. Linton (1966), Autonomous equational categories. J. Math. Mech. 15, 637–642.
V.R. Pratt (1993a), Linear Logic for Generalized Quantum Mechanics. In Proceedings
Workshop on Physics and Computation, Dallas, IEEE Computer Society.
V.R. Pratt (1993b), The Second Calculus of Binary Relations.
MFCS’93, Gdańsk, Poland, 142–155.
In Proceedings of
V.R. Pratt (1995), The Stone Gamut: A Coordinatization of Mathematics. In Proceedings
of the conference Logic in Computer Science IEEE Computer Society.
Dept. of Math. and Stats.
McGill University
805 Sherbrooke St. W
Montreal, QC H3A 2K6
Email: [email protected]
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